Hybrid ballistic–diffusive solution to the frequency-dependent phonon Boltzmann Transport Equation

It is commonly assumed that the distribution of molecular velocities follows a Maxwellian distribution also in the range of high and ultrahigh vacuum, i.e., at large Knudsen numbers. This distribution is theoretically derived from statistical analysis of an ensemble of molecules with frequent molecule–molecule collisions. Such a situation prevails at higher pressures, i.e., at small Knudsen numbers. However, at smaller pressures the molecule–wall collisions prevail, and due to the different nature of these collisions, the statistical arguments fail. As a consequence, the molecular velocity distribution cannot be predicted a priori at such pressures. In the present paper the applicability of the Maxwellian distribution in vacuum metrology at large Knudsen numbers, i.e., in the molecular regime is investigated. For this purpose, the pressures generated in primary standards of the static and dynamic expansion type are calculated for an arbitrary velocity distribution after expansion. The pressure generated by static expansion is proportional to the expectation value 〈v2z〉, that generated by dynamic expansion proportional to the ratio of expectation values 〈v2z〉/〈‖vz‖〉 (vz denotes the one-dimensional velocity component). Available experimental calibration data obtained by static and dynamic expansion of common gases are compared to each other and with direct pressure measurements with a liquid column manometer. These data provide a decisive test for the expectation values 〈v2z〉 and 〈‖vz‖〉 in the molecular range. Excellent agreement within the uncertainty of less than 1% between experimental values and values calculated assuming a Maxwellian distribution is found. Thus one may conclude that in common vacuum metrology the Maxwellian distribution can be applied for deriving pressures also in the molecular regime with high accuracy.

The problems of the gas dynamic interactions of two spherical particles and a single spherical particle with a plane wall are reviewed here. The resistance and mobility functions that describe the relationship between the forces and torques acting on the spheres with their translational and angular velocities are summarized for continuum and near-continuum (small Knudsen number) flows. For intermediate and large Knudsen number (free-molecule) flows, these functions are only known in their asymptotic limit of infinitely large interparticle or particle-wall separation distances (isolated particles). Potentially useful analytical and computational methods to treat these regimes are also reviewed.

Nondiffusive phonon transport presents significant challenges in micro/nanoscale thermal characterization, compounded by the limitations of experimental-numerical techniques and the presence of measurement noise. Additionally, inverse modeling and uncertainty quantification (UQ) for submicron thermal transport remain under-explored. In this study, we introduce a physics-informed Bayesian deep learning framework designed to address phonon Boltzmann transport equation (BTE)-based forward and inverse problems leveraging limited and noisy data. Our approach combines Bayesian neural networks with a nonparametric variational inference method, formulating the BTE-constrained training in a Bayesian manner. This enables the estimation of the posterior distribution of neural network parameters and unknown equation parameters based on a likelihood function that incorporates uncertainties from both the measurement data and the BTE model. Through numerical experiments on various phonon transport scenarios, we demonstrate that our method can accurately reconstruct temperature and heat flux profiles, infer critical quantities of interest (e.g., Knudsen number), and provide robust uncertainty quantification, even when data is sparse and noisy. This framework enhances our capability to conduct nondiffusive thermal simulations and inverse modeling with quantified uncertainty, offering a powerful tool for advancing thermal transport research and optimization in micro/nanoscale devices.

Direct Simulation Monte Carlo (DSMC) solves the Boltzmann equation with large Knudsen number. The Boltzmann equation generally consists of three terms: the force term, the diffusion term and the collision term. While the first two terms of the Boltzmann equation can be discretized by numerical methods such as the finite volume method, the third term can be approximated by DSMC, and DSMC simulates the physical behaviors of gas molecules. However, because of the low sampling efficiency of Monte Carlo Simulation in DSMC, this part usually occupies large portion of computational costs to solve the Boltzmann equation. In this paper, by Markov Chain Monte Carlo (MCMC) and multicore programming, we develop Direct Simulation Multi-Chain Markov Chain Monte Carlo (DSMC3): a fast solver to calculate the numerical solution for the Boltzmann equation. Computational results show that DSMC3 is significantly faster than the conventional method DSMC.

In this work we use a three dimensional Molecular Dynamics simulation method to study the effect of different geometries and Knudsen number regimes on the gas flow in micro-nanochannels. Argon molecules have been used for the simulations. Thermal wall and diffusive-specular wall types were used for the boundaries of the channels. The velocity profiles in the channel were obtained and analyzed with three different channel geometries that are commonly used in the industry: circular, rectangular (square), and slit channel. We found that when using the same driving force, the maximum velocity of the flow increases when the geometry changes in the order from circular geometry to rectangular geometry to slit geometry, where the latter becomes 1.2∼1.5 times as large compared with either the rectangular or circular channel. While the absolute values of the velocity profiles show a distinct difference according to the different geometries, geometry effect on the shape of the velocity profile also shows interesting features. Rectangular tube shows much flatter profile compared with the other two channels. Also the effect of the size of the channels and different Knudsen numbers on the velocity profiles is investigated. Two different sizes were used here: 100nm and 10nm corresponding to typical sizes of a nano channel and carbon nanotubes. We found that the Knudsen number has an effect on the slip and maximum flow velocity for the slit geometry even for higher Knudsen number. For the Kn higher than approximately 3, it was found that the Knudsen number has a small influence on the slip flow velocity for the circular channel and rectangular channel than for lower Knudsen number.

Ultra-efficient and parameter-free computation of submicron thermal transport with phonon Boltzmann transport equation

A mean free path kinetic theory of gaseous void transport with simultaneous surface diffusion on the pore walls has been developed for diffusion through a porous medium. A variational upper bound expression for the effective diffusivity is applied to a model porous medium, a bed of randomly placed, freely overlapping solid spheres all of the same radius. The effects of tortuosity are rigorously considered by explicitly including the flux of diffusing material around obstructions in its path in the trial functions. Numerical calculations of the variational integrals are performed for porosities from 0.01 to 0.90 and the full range of Knudsen numbers and surface diffusivities. From asymptotic values at large Knudsen numbers, a table of effective diffusivities versus the surface diffusion coefficient is presented for pure Knudsen diffusion in the voids.

In this paper we investigate a two-dimensional dilute granular flow around an immersed cylinder using discrete element computer simulations. Simulation measurements of the drag force acting on the cylinder, Fd, are expressed in terms of a dimensionless drag coefficient, Cd=Fd/[12ρν∞U∞2(D+d)], where ρ is the upstream particle mass density, ν∞ is the upstream solid fraction, U∞ is the upstream velocity, and (D+d) is the sum of the cylinder diameter, D, and surrounding particle diameter, d. The drag coefficient increases rapidly with decreasing Mach number for subsonic Mach numbers, but remains insensitive to Mach number for supersonic values. The drag coefficient is also a strong function of the flow Knudsen number, with the drag coefficient increasing with increasing Knudsen number and approaching an asymptotic value for very large Knudsen numbers. The drag coefficient decreases with decreasing normal coefficient of restitution and is relatively insensitive to the friction coefficient. Bow shock structures and expansion fans are also observed in the simulations and are compared to similar structures observed in compressible gas flows.

In this study, the grand (translational) friction tensor for free-molecule (large Knudsen number) flows over two, unequal-size cylinders is obtained analytically considering both ‘‘shielding’’ effects and multiple collisions off cylinder surfaces, as well as including both diffuse and specular molecule-surface boundary conditions. A perturbation solution is used where the leading-order term accounts for molecular collisions from the free-streaming gas including all shielding effects. The first-order correction term is also obtained here and accounts for collisions by molecules reflected off neighboring surfaces from the free-stream gas. It is shown that the grand friction tensor is symmetric under the conditions of small ratios of the free-stream gas velocity to the thermal gas velocity, the same type of molecular reflection boundary condition on both cylinders, and constant temperature. It is also shown that the two-term expansion formula is accurate under isothermal conditions in the so-called ‘‘far-field’’ limit where the surface to surface cylinder separation distance is greater than approximately one to two cylinder diameters for equal-size cylinders. The accuracy of the two-term expansion is also shown to diminish as the two cylinders approach one another (‘‘near field’’ behavior). It is also demonstrated how the far-field results given here can be extended to more than two cylinders and to cylinders moving relative to the free-stream gas. This study compliments an earlier study [R. Ying and M. H. Peters, J. Fluid Mech. 207, 353 (1989)] for small Knudsen numbers, or the ‘‘slightly rarefied gas’’ case.

Phonon Boltzmann Transport Equation based modeling of time domain thermo-reflectance experiments

A hybrid particle scheme, which is based on the direct simulation Monte Carlo (DSMC) method and incorporates a low diffusion particle method for continuum flow simulation, is extended for improved efficiency and wider applicability. The hybrid scheme is intended for simulation of gas flows involving a wide range of Knudsen number regimes, where strong coupling is desired between calculations in rarefied and continuum flowfield regions. A series of modifications are presented for cell-based numerical weight and time step adaptation, diffusive transport of nonequilibrium internal energy modes, and decreased sensitivity to DSMC cell size. Proposed modifications are employed in a hybrid simulation of a Mach 20 flow around a cylinder with a global Knudsen number of 0.002, for which a factor of 2.6 efficiency increase relative to full DSMC computation is demonstrated. I. Introduction n a variety of gas flow problems, portions of the flowfield exhibit large translational nonequilibrium effects and require relatively expensive simulation techniques based on the Boltzmann equation, while other regions are within low Knudsen number (Kn) continuum flow regimes and can be prohibitively expensive to simulate using such Boltzmann equation techniques. These “multiscale” flow problems include flows around atmospheric entry vehicles and hypersonic aircraft, where local Kn values based on gradient length scales may be very small through much of the flowfield but become large within shock, forebody boundary layer and wake regions. Other examples include rocket exhaust plumes and fuel venting flows at high altitude, where the mean free path may differ by several orders of magnitude between nearfield and farfield regions. A. Simulation Methods for Multiscale Gas Flows Accurate simulations may be performed for some of these flows using a two step, one-way coupled approach. 1-3 In this type of approach, continuum and rarefied regions are considered independently and different simulation techniques are applied in each region. In general, however, effects of two-way coupling between rarefied and continuum regions must be considered, and a single simulation employing a coupled hybrid algorithm is required. In one popular type of hybrid algorithm, a computational fluid dynamics (CFD) technique based on the Navier-Stokes equations is used in low-Kn continuum regions, the direct simulation Monte Carlo (DSMC) method is employed in rarefied regions, and additional procedures are used for evaluation of continuum breakdown and two-way information transfer between CFD and DSMC calculations.

Nonisothermal condensation on spherical aerosol particles from the grad solution of the boltzmann equation

In this paper, we present a model that combines lattice dynamics and the phonon Boltzmann transport equation (BTE) to analyze strain effect on the cross-plane phonon thermal conductivity of silicon wire-germanium host nanocomposites. For a given strain condition, mechanical strain is translated to crystal lattice deformation by using the Cauchy–Born rule. Strain-dependent phonon thermal properties of Si and Ge obtained from lattice dynamics with Tersoff empirical interatomic potential are then incorporated into the BTE, in which ballistic transport within one material and diffuse scattering between Si–Ge interface are employed. The strain-dependent BTE is solved numerically on an unstructured triangular mesh by using a finite volume method. Nanocomposites with different Si nanowire cross sections are also investigated. The results show that the phonon thermal conductivity of the nanocomposites can be significantly decreased (or increased) by a tensile (or compressive) strain. With the same length change, hydrostatic strain produces a larger variation in phonon thermal conductivity than uniaxial strain. In addition, it is shown that with the same atomic percentage, the cross-sectional shape makes little difference to the thermal conductivity except at very small characteristic lengths of the Si nanowire.

In compact transistors, large electric fields near the drain side create hot spots whose dimensions are smaller than the phonon mean free path in the medium. In this paper, we present a study of unsteady hot spot behavior. The unsteady gray phonon Boltzmann transport equation (BTE) is solved in the relaxation time approximation using a finite volume method. Electron-phonon interaction is represented as a heat source term in the phonon BTE. The evolution of the temperature profile is governed by the interaction of four competing time scales: the phonon residence time in the hot spot and in the domain, the duration of the energy source, and the phonon relaxation time. The influence of these time scales on the temperature is investigated. Both boundary scattering and heat source localization effects are observed to have considerable impact on the thermal predictions. Comparison of BTE solutions with conventional Fourier diffusion analysis reveals significant discrepancies.

We study the pressure-driven steady gas flow, imposed by temperature or density gradients, over a backward-facing step in a two-dimensional microchannel. Focusing on the near-free-molecular regime of high Knudsen (Kn) numbers, the problem is analyzed asymptotically based on the Bhatnagar, Gross and Krook kinetic model, and supported by numerical Discrete Velocity Method and Direct Simulation Monte Carlo calculations. The wall conditions are formulated using the Maxwell model, superposing specular and diffuse surface conditions. The asymptotic solution contains the leading-order free-molecular description and a first-order integral representation of the near-free-molecular correction. Our results indicate that flow separation at the step can occur at arbitrarily large (yet finite) Knudsen numbers in channels with specular surfaces (i.e., having an accommodation coefficient of α=0), driven by temperature differences between the inlet and outlet reservoirs. It is then shown that detachment is significantly suppressed by density variations between reservoirs and partially diffuse surfaces (with α≳0.3). While the mass flow rate in a specular channel decreases with decreasing Kn≫1 in a density-driven setup (in line with the Knudsen Paradox), it increases in a temperature-driven flow. The results are obtained for arbitrary differences between the inlet and outlet reservoir equilibrium properties, and are rationalized using the linearized problem formulation.